1. Field of the Invention
This patent specification relates to gamma ray measurements in subterranean formations. More particularly, this patent specification relates to systems and methods for forward modeling for gammy ray measurement analysis of subterranean formations.
2. Background of the Invention
In nuclear logging oilfield applications, Monte Carlo methods are preferred for their accuracy. However, in practice many Monte Carlo methods are unsuitable for real-time log analysis due to the limited computational speed of modern computers.
An example of a conventional linear based forward modeling technique is described in Charles C. Watson, “Monte Carlo Computation of Differential Sensitivity Functions”, Trans. Am. Nucl. Soc., vol. 46, page 655, 1984, and Charles C. Watson, “A Spatial Sensitivity Analysis Technique for Neutron and Gamma-Ray Measurements”, Trans. Am. Nucl. Soc., vol. 65 (Suppl.1), pp. 3-4, 1992 both of which are incorporated herein by reference, and hereinafter referred to as “the Watson Papers.” By modeling the dominant gamma-ray interactions of Compton scattering and photoelectric absorption, this technique can be used to predict the detector response of a density logging tool. The primary advantage of the density sensitivity method is its very fast computational speed, in that it can provide answers on a sub-second scale. Its basic premise assumes a linear relationship between the detector response and changes in the density and Pe properties of the formation. The space around the tool is divided into a grid of cells, each of which is assigned a sensitivity. The contribution of each cell for the response estimate comes from pre-calculated spatial sensitivity maps. Further details of the density sensitivity function technique are disclosed in U.S. Pat. No. 5,334,833, incorporated herein by reference. The performance of this linear technique is modest, with relative accuracies of a few percent in count space which must then be converted to density space. For example, when applying the method to the LWD Vision 475 density tool from Schlumberger, the modeling error in comparison to experimental data was found to be as much as 0.1 g/cc within 1″ water standoff when covering typical spatial variations in density from 1 to 3 g/cc. The limited nature of the 1st-order method is also apparent in that the density sensitivity functions are not identical when calculated using different reference formations. Some improvement in accuracy can be realized by modifying the sensitivity functions on a case-by-case basis, but such adjustments are not fully general. A. Mendoza, C. Torres-Verdin, and W. Preeg, “Rapid Simulation of Borehole Nuclear Measurements With Approximate Spatial Flux-Scattering Functions”, SPWLA 48th Annual Logging Symposium, Jun. 3-6, 2007 proposes a spatial flux-scattering functions (FSFs) technique to rapidly simulate neutron and (gamma-gamma) density porosity logs. This technique is very similar to Watson's sensitivity technique described above. As a result, the speed of the FSF model is comparable to Watson's model. However, the accuracy achieved by the FSF model is only 10%, which is not a significant improvement over the older models.